Method and apparatus for controlling diameter of a silicon crystal ingot in a growth process

ABSTRACT

An improvement to a method and an apparatus for growing a monocrystalline silicon ingot from silicon melt according to the CZ process. The improvement performs defining an error between a target taper of a meniscus and a measured taper, and translating the taper error into a feedback adjustment to a pull-speed of the silicon ingot. The conventional control model for controlling the CZ process relies on linear control (PID) controlling a non-linear system of quadratic relationship defined in the time domain between the diameter and the pull-speed. The present invention transforms the quadratic relationship in the time domain between the diameter and the pull-speed into a simile, linear relationship in the length domain between a meniscus taper of the ingot and the pull-speed. The present invention applies a linear control (modified PID) which operates in the length domain, and controls a system that has a linear relationship between the ingot taper and the pull-speed in the length domain to control the diameter of a growing silicon ingot.

BACKGROUND

This invention relates generally to improvements in controlling thegrowth process of a monocrystalline silicon ingot and, particularly, toa method and apparatus for accurately controlling the diameter of amonocrystalline silicon ingot during its growth process.

The Czochralski (CZ) process is used to obtain monocrystals. The mostimportant application thereof is growing a monocrystalline siliconingot, which is sliced into silicon wafers for fabrication ofsemiconductor circuits thereon. Briefly described, the CZ processincludes melting a charge of polycrystalline silicon in a quartzcrucible and lifting a monocrystalline seed from the surface of the meltsilicon. As the seed is being lifted from the melted silicon,monocrystalline silicon grows from the seed and forms a cylindricalingot. The modern CZ process produces silicon ingots having a diameteras large as 300 mm.

The key to produce silicon wafers with a uniform diameter is to producesilicon ingots with a uniform diameter along the length. It is wellknown by those skilled in the art that an increase in the pull-speed ofthe seed results in a reduction of the diameter of a growing siliconingot therefrom and vise versa. It is also well known that an increasein the temperature of the silicon melt in the crucible results in areduction of the diameter of a growing silicon ingot and vise versa.While diameter control sounds simple, it requires a sophisticatedfeedback control.

Conventionally, the CZ process is performed with the PID(proportional-integral-derivative) control method to control thediameter of a growing silicon ingot. The PID controller receives anerror signal representing a difference between the target or desireddiameter of a growing silicon ingot and the diameter of the siliconingot actually observed. The PID controller then processes the deviationof the diameter as a function of time and transforms it into apull-speed error. The pull-speed error is used to adjust the pull-speedof the seed.

Pull-speed control alone is usually insufficient to control the diameterof a growing silicon ingot satisfactorily. Thus, the CZ process isperformed with additional PID controller specifically designed tocontrol the temperature of the silicon melt in the crucible. The abovepull-speed error is integrated over time to derive a temperature error.The derived temperature error, the target temperature from thetemperature profile and the temperature actually measured are summed andprovided to the second PID controller to adjust the temperature of thesilicon melt.

Although the above described CZ process using two PID controllers toadjust both the pull-speed and the melt temperature simultaneously iswidely used in producing silicon ingots, further improvements are neededto produce silicon ingots with diameters sufficiently satisfactorilyuniform. In these days, the required standard for precisely andaccurately controlling the intrinsic properties of silicon ingots duringtheir growth has become much higher and stricter than it used to be. Itis well known that variations of the pull speed that are performed tocontrol the ingot diameter have a negative effect on a defectdistribution within the ingot. It is further well known that pull speedvariations have a negative effect on morphological stability during thegrowth of heavily doped ingots. It is therefore necessary to minimizethe pull speed variations when controlling the diameter of the siliconingot during the growth thereof.

Concerning the diameter and growth control, there are three categoriesof error sources that all lead to pull speed variations while growingsilicon ingots to have them with a uniform diameter. The first categoryof error is caused by temperature fluctuations in the melt. It is wellknown that the temperature fluctuations in the melt are caused bybuoyancy effects which bring about turbulences in the melt flow. Suchtemperature fluctuations cause changes of the crystallization rate, andsuch crystallization rate changes then cause changes of the ingotdiameter. The diameter control system is designed to react to thesediameter changes by outputting pull speed adjustments which result inpull-speed variations.

It is generally known that the melt flow turbulences can be reduced byapplying a magnetic field which functions to reduce the melt temperaturefluctuations and thereby reduce the pull speed adjustments by thediameter control system. The diameter deviations caused by thetemperature fluctuations in the melt are more significant in smalldiameter ingots than in large diameter ingots, since the temperaturefluctuations are localized in the melt, and the effects thereof areaveraged over the cross section of an ingot if the diameter thereof islarge. However, even under the condition that a large diameter ingot isgrown within a magnetic field, there is still need for reducing the pullspeed variations.

The second error source resides in the diameter feedback control itselfand is caused by an inferior control model. Diameter control iscustomarily performed, using a PID controller. Those skilled in the artof control theory know that PID controllers are perfect for use incontrolling systems that are governed by a linear differential equationof up to 2^(nd) order. To some degree, PID control can also be used forcontrolling non-linear or higher order systems, but only in cases wherethe control performance and stability are not so important. When it isrequired to deal with systems that follow non-linear or higher orderequations under the condition that the high control performance and highstability are needed, a specialized controller needs to be developed.However, because being convenient and also widely used, the conventionalPID controllers are still used in the ingot diameter control, despitethe fact that the required standard for control stability, or forreducing pull speed variations, is highly strict in these days andcannot be met by the conventional PID controllers.

The third source of error is caused by input errors, such as noise in aninput signal of a diameter measurement indicative of the diameter of agrowing ingot. Such noise in the input signal directly affects thediameter control system, causing unnecessary pull-speed variations. Anerror of this kind may not be so obvious in the prior art controlsystems because the prior art control systems usually suffer errorsattributed to inferior control models adopted therein and such errorsare large enough to dominate the input errors. However, an impact of theinput errors on the control stability becomes obvious when a specializedcontrol system is used in which an error from an adopted control modelis small.

SUMMARY OF THE INVENTION

The present invention specifically addresses errors from the seconderror source and provides a method and an apparatus for producingsilicon crystal ingots with a uniform diameter along their lengths.

Diameter control is nowadays implemented by a digital computer runningon a software algorithm, which is patterned after the analog PID controlthat had existed long before the digital computer appeared. As discussedabove, PID control is perfect for use in controlling systems which aregoverned by a linear differential equation up to order 2. The PIDcontroller can also be used in controlling systems which are governed bya non-linear equation or a linear differential equation of more thanorder 2, however, at the cost of control performance and stability.

Unfortunately, it is not correctly understood exactly how the ingotdiameter responds to a change of the pull-speed. A general perception isthat a change of the ingot diameter is the direct result from a changeof the pull-speed. Hence, in the prior art diameter control, the P-termis the dominant factor of control in which a change of the pull-speed iscalculated as being proportional to a diameter error. However, this israther an oversimplification of the relationship between the diameterand the pull-speed. What actually happens is that the meniscus appearingbetween the solid ingot and the liquid silicon melt changes its heightas a direct response to a pull-speed change for correcting a pull-speederror.

The pull-speed error is a difference between the zero taper pull-speedand an actual pull-speed observed. The zero taper pull-speed here meansthe pull speed at which the crystal grows cylindrical. The zero taperpull-speed is determined by the melt temperature and changes thereof. Atthe zero taper pull-speed, the meniscus can maintain the correct heightif the melt temperature is constant. When the actual pull-speed deviatesfrom the zero taper pull-speed, so does the meniscus height from thecorrect height. Such a deviation of the meniscus height results in achange of the wetting angle at the 3-phase boundary, causing the ingotto grow at a taper angle, instead of growing perfectly vertically orcylindrically. Unless the diameter error becomes large enough to modifythe thermal conditions, the ingot would continue to grow whilepreserving the taper angle. If the diameter error could become largeenough, it would eventually find a new equilibrium diameter with whichthe ingot will grow cylindrically. However, if the temperature wasabsolutely homogeneous in the crucible, e.g. if the temperature wouldn'tchange with the radius of growing ingot, the ingot diameter would justcontinue to grow at the taper angle that is determined by the pull-speederror. Thus, although it is true that a change of the pull-speed causesa change of the diameter, it is a misunderstanding to think that thediameter can be controlled directly by changing the pull-speed. A changeof the diameter is not at all the direct result from a change of thepull-speed.

Importantly, the ingot diameter changes over time at a rate which isproportional to a taper multiplied by the actual pull-speed. Since thetaper is approximately proportional to a pull-speed error, a diameterchange per time is therefore approximately proportional to a termproportional to the pull-speed error plus a term proportional to thesquare of the pull-speed error. Even though this is still rather asimple model, the model is very useful to design a new diameter controlsystem which can significantly reduce pull-speed variations. First ofall, the model illustrates the cause of the errors in the prior artcontrol model, i.e., the non-linear term, or the quadratic pull-speedterm, contained in the control model. It further illustrates that it isnot the diameter but the diameter taper which is approximatelyproportional to the pull-speed error.

More importantly, it also shows that it is not very accurate tocalculate the integral term, using a diameter error, as performed by theprior art PID control. Again, unless the diameter error is allowed tobecome so large as to affect the thermal conditions, it is not a goodmeasure for use in deriving an automatic pull-speed offset (the integralterm). Rather, it is the taper error that derives a pull-speed offset,because the taper error relates to the meniscus height error, whichrelates to the pull-speed error, which relates to the melt temperature.Accordingly, the taper error should be used to derive the automaticpull-speed offset by integration.

Further, the calculated value from the integral term which is derivedfrom the taper error is an estimate for the pull-speed for achieving themomentary zero-taper growth (cylindrical growth) and thus also a betterrepresentation of a temperature error that causes a deviation of thezero-taper pull-speed from its target.

In order to eliminate the control error attributed to the quadratic termin the differential equation governing the diameter-pull-speed dynamic,the inventor of the present application has transformed the quadraticrelationship in the time domain between the diameter change per time andthe pull-speed, which is expressed by the following equation:

${{\left. \frac{\partial{r(t)}}{\partial t} \right.\sim{v(t)}} \cdot \Delta}\;{v(t)}$into a simple, linear relationship in the length domain between thediameter change per length (the diameter taper) of the ingot and thepull-speed, which is expressed by the following equation:

${\left. \frac{\partial{r(z)}}{\partial z} \right.\sim\Delta}\;{{v(z)}.}$Based on that, a new simple PID-controller becomes possible whichoperates on a linear system in the length domain, rather than operatingon a non-linear system in the time domain. Thereby, it is possible toeliminate the biggest control model error without the need to develop acomplicated non-linear control.

More specifically, the present invention provides a method for growing amonocrystalline silicon ingot from silicon melt according to the CZprocess. To implement a feedback control based on the linearrelationship between the meniscus taper and the pull-speed, the methodcomprises defining an error between a target taper of a meniscus and ameasured taper, and then translating the taper error into a feedbackadjustment to the pull-speed of the silicon ingot. Since control isimplemented based on the simple linear relationship, the presentinvention can achieve the accurate feedback control of the diameter ofthe silicon ingot. Being able to use the well-known and robust PIDmechanism, the present invention is very convenient and well suited forindustrial application.

Another aspect of the present invention also addresses the problem thatthe standard PID controller increases, without limits, its output inproportional relation to the input error. The controller according tothe present invention defines slope limits to make sure that the slopeat which the controller steers back to the target diameter never exceedsthe predetermined limits, even with a very large diameter error. Thus,the controller according to the present invention can avoid too drasticchanges of the growth interface when correcting a large diameter error.

In the present invention, the target taper may be derived from thediameter and the pull-speed by processing these data in a kinematicmodel filter or a more elaborate tracking filter. Thus, the methodaccording to the present invention may further comprise defining anerror between a target diameter of the silicon ingot and a measureddiameter, and translating the diameter error into the target taper atwhich the system intends to steer back to the target diameter.Translating the diameter error into the target taper may be achieved bymultiplying the diameter error with a constant.

The feedback control according to the present invention is unique inthat the taper-errors are integrated over the pulled ingot length toderive an i-term of the feedback adjustment, instead of having thediameter errors integrated over time as performed in the conventionaldiameter control. The feedback operation also multiplies the taper errorwith a constant to derive a p-term of the feedback adjustment.

In addition to the feedback control, the present invention alsoimplements a feed-forward control. The feed-forward control according tothe present invention predicts an adjustment to the pull-speed from thetarget taper. Thus, the method of the present invention furthercomprises the operation of translating the target taper into afeed-forward adjustment to the pull-speed. The adjustment from thefeed-forward control and the adjustment from the feedback control mayboth be used to adjust the pull-speed. The adjustment from thefeed-forward control enables quick response to a diameter deviation.

The integral term of the controller yields a good estimate of the zerodiameter error pull-speed or the zero taper pull-speed, even when it isoperating under target taper clipping conditions. In other words, thedifference between the target pull-speed and the i-term adjustment fromthe controller is a perfect input for the heater control that containsless components that are unrelated to the melt temperature. This isparticularly helpful while correcting a large diameter error.

The present invention implements temperature control of the siliconmelt. For this purpose, the method according to the present inventionfurther comprises the operation of translating the i-term adjustment tothe pull-speed into a deviation of the temperature of the silicon melt.The i-term adjustment is a good representation of a temperaturedeviation and is integrated over time to derive the temperaturedeviation. Using the temperature deviation, the method according to thepresent invention effects a PID control to control the temperature ofthe silicon melt.

The present invention is contemplated to be implemented in a modifiedcontroller in which a PI-control operates in the length domain with adynamic taper set-point, or a target taper, derived from a diameterdeviation. Such a modified controller has advantages of improved controlstability and high adaptation for tuning. Please note, however, that thepresent invention is also implementable in a regular PID controlleroperating in the length domain. Such a regular PID controller operatingin the length domain has already been found achieving a significantimprovement over prior art.

Therefore, the present invention also provides a method for growing amonocrystalline silicon ingot from silicon melt according to the CZprocess in which an error is defined between a target value of aparameter and a measured value of the parameter and then integrated in alength domain to derive a feedback adjustment to a pull-speed of thesilicon ingot in order to make a diameter of the silicon ingot uniformalong its length.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic view showing a silicon crystal growing apparatusaccording to the present invention.

FIG. 2 is a diagram showing a control model according to the presentinvention which is used in the apparatus shown in FIG. 1

FIG. 3 is a schematic view showing a meniscus formed at the interfacebetween a growing silicon ingot and silicon melt.

FIG. 4 is a graph showing experimental data illustrating therelationship between a change of the diameter of a growing silicon ingotand a change of the pull-speed.

FIG. 5 is a graph comparing traces of pull-speed control performed bythe present invention and the conventional PID control.

FIG. 6 is a graph showing control of the diameter and the pull speedunder the present invention.

FIG. 7 is a graph showing control of the diameter and the pull speedunder the conventional PID control.

DETAILED DESCRIPTION OF THE DRAWINGS AND THE PRESENTLY PREFERREDEMBODIMENTS

Hereinafter, a detailed explanation of the method and apparatus forcontrolling the diameter of a monocrystalline silicon ingot according tothe present invention will be given with reference to the attacheddrawings.

Referring now to FIG. 1, an apparatus according to the present inventionis shown for use in implementing a CZ crystal growing process. In thisfigure, the apparatus comprises a furnace 1. Inside the furnace 1, aquartz crucible 2 is provided which holds silicon melt 6. The quartzcrucible 2 is received by a graphite susceptor 3, which is fixed at thetop of a drive shaft 4. The drive shaft 4 moves vertically and rotatesto thereby move the quartz crucible 2 vertically and also rotate it. Acarbon heater 7 surrounds the susceptor 3 and heats the crucible 2 tocontrol the temperature of the silicon melt 6 in the crucible 2. Aninsulation tube 8 is placed between the heater 7 and the furnace wall.

A tubular radiation heat shield 11 is suspended above the silicon melt6. The heat shield 6 prevents changes of the heat history of a siliconingot being lifted and also prevents impurities, such as CO gas from theheater 7, from entering the silicon ingot being lifted. A water cooledcooling pipe 10 is attached to the inside of a neck 14 of the furnace 1.The cooling pipe 10 controls the heat history of the silicon ingotduring the lifting process. Between the cooling pipe 10 and the innerwall of the neck 14, an annular gas passage is formed through which afeed pipe 20 supplies Argon gas inside the furnace 1.

A wire 9 runs vertically through the neck 14 and the cooling pipe 10down to the silicon melt 6 in the crucible 2. The wire 9 holds amonocrystalline silicon seed at its end. A servo motor 20 pulls up thewire 9 and lifts the crystal seed from the surface of the silicon melt6. FIG. 1 shows a silicon ingot 5 growing out of the silicon melt 6 asthe servo motor 20 pulls the wire 9 up. The wire 9 and thus the seed canbe rotated by another motor not shown in the figure.

Windows 12 and 13 are formed in a shoulder of the furnace 1. An ADRsensor 15 measures through the window 12 the diameter of the siliconingot 5 being pulled up. A line camera 16 is used to observe through thewindow 13 the crystal growing process occurring inside the furnace 1. Awindow 17 is formed in the side wall of the furnace 1, through which anATC sensor 18 monitors the temperature of the heather 7. Further, adichromatic thermometer 19 is placed at the top of the furnace 1 andmeasures the temperature of the silicon melt 6.

All the data collected by sensors 15, 16, 18 and 19 are supplied to acontroller 22. The controller 22 processes the supplied data andcontrols the servo motor 20 and the heather 7 so as to produce a siliconingot with a uniform diameter along the length. The controller 22 alsocontrols a vertical movement and a rotation of the drive shaft 4.Particularly, the controller 22 controls a vertical movement of thedrive shaft 4 to keep constant the vertical position of the interfacebetween the growing silicon ingot 5 and the silicon melt 6.

Referring to FIG. 1, how the CZ process is implemented inside thefurnace 1 will be explained. A charge of polycrystalline silicon isfirst placed in the crucible 2. Argon gas is flown into the furnace 1through the feed pipe 20 to fill the furnace 1 with the argon gas. Theheather 7 is turned on to heat the crucible 2 and melt thepolycrystalline silicon inside the crucible. The heather 7 is controlledto maintain the temperature of the silicon melt 6 according to apredetermined temperature profile.

Next, the servo motor 20 is driven to lower the wire 9 until themonocrystalline silicon seed attached to the end of the wire 9 ispartially submerged in the silicon melt 6. Thereafter, the quartzcrucible 2 and the crystal seed start begin rotated in the oppositedirections. The servo motor 20 then begins pulling up the wire 9according to a predetermined pull-speed profile. As the crystal seed isbeing pulled up, the ingot 5 grows from the seed.

At regular intervals, e.g., every one second, the controller 22 collectsthe diameter information from the sensor 15 and the temperatureinformation from the sensor 19. Based on the collected information, thecontroller controls the servo motor 19 and the heather 7. By controllingthe pull-speed and the temperature of the silicon melt 6, the diameterof the ingot 5 gradually increases from the crystal seed to thereby forma conical neck portion. When the ingot has grown to the intendeddiameter, the controller shifts its control so that the growing ingot 5will have the constant diameter. When the ingot 5 has grown to theintended length, the controller 22 again shifts its control to graduallyreduce the diameter of the ingot 5 to form a conical tail portion.

During the ingot growing process, as the ingot 5 grows, the surface ofthe silicon melt goes lower. To compensate the decent of the meltsurface and keep constant the vertical level of the interface betweenthe ingot 5 and the silicon melt 6, the controller has the crucible 2raised by the drive shaft 4. When the diameter of the tail portionbecomes nearly zero, the ingot 5 is lifted away from the silicon melt.The heater 7 is turned off to terminate the CZ process.

FIG. 2 illustrates the control model of the present invention forproducing an ingot with a uniform diameter along the length. In FIG. 2,a process table 100 is shown which is stored in the controller 22. Theprocess table includes information of the intended ingot length 101. Theprocess table 100 also includes a pull-speed profile table 102 whichincludes information on the target pull-speeds to be achieved during theCZ process. A diameter profile table 103 includes information on thetarget diameters to be achieved during the CZ process. A temperatureprofile table 104 includes information on the target temperatures to beachieved during the process.

Shown on the other side of FIG. 2 is the furnace 1. The servo motor 19for pulling up the wire 9 and the heater 7 for heating the crucible 2are shown inside the furnace 1. Control signals are supplied to theservo motor 19 and the heather 7 to achieve the desired pull-peed andtemperature of the silicon melt in order to produce a silicon crystalingot with a uniform diameter along the length.

The control model used in the present invention operates not in the timedomain but uniquely in the length domain. FIG. 3 shows the interfacebetween the growing ingot 5 and the silicon melt 6. Between the solidingot 5 and the liquid melt 6, a tapered portion, called a meniscus, isformed. Those skilled in the art recognize that the shape of themeniscus is an important determinant to the diameter of the growingcrystal ingot. In the figure, “α” represents an angle of the meniscustaper and “h” represents a height of the meniscus.

Given the shape of the meniscus, the diameter of a growing ingot can beexpressed by following Equation (1):

$\begin{matrix}{\frac{\partial r}{\partial t} \approx {v \cdot {\Delta\alpha}}} & (1)\end{matrix}$The equation tells that the larger the pull-speed becomes, the larger adiameter change per time becomes. Also, a diameter change per time isproportional to a deviation of the meniscus wetting angle from the angleat which the ingot grows perfectly vertically or cylindrically. Thoseskilled in the art also recognize that the meniscus angle deviation hasan approximate linear relationship with a change of the pull-speed.Δα≈const·Δv  (2)The relationship between the diameter and the pull-speed is derived fromEquations (1) and (2) as follows:

$\begin{matrix}{\frac{\partial r}{\partial t} = {{{const} \cdot v \cdot \Delta}\; v}} & (3)\end{matrix}$Equation (3) describes the system that is to be controlled by thediameter controller.

Equation (3) shows that the diameter and the pull-speed have a quadraticrelationship in the time domain. PID control cannot achieve optimalcontrol on such non-linear system, because PID control by design canonly optimally compensate errors of systems governed by a lineardifferential equation of up to second order.

In the CZ process, a pull-speed determines the growing speed of an ingotat the meniscus. Thus, a pull-speed is a change of vertical location (z)of an ingot at the meniscus.

$\begin{matrix}{v = \frac{\partial z}{\partial t}} & (4)\end{matrix}$Equations (3) and (4) yield the follow equation:

$\begin{matrix}{\frac{\partial r}{\partial z} = {{{const} \cdot \Delta}\; v}} & (5)\end{matrix}$Equation (5) shows that a diameter change per pulled length has anapproximately linear relationship with a change of the pull-speed. Achange of diameter per pulled length means a taper of the ingot. Thus,Equation (5) tells that the ingot taper has an approximately linearrelationship with a change of the pull-speed. Equation (5) is intuitivethat the diameter and the pull-speed have a linear relationship in thelength domain. Therefore, by applying the PID control in the lengthdomain, rather than the time domain, it becomes possible to achieveoptimal control for the ingot growing process, without sacrificingcontrol performance and stability.

However, since the control computer still samples inputs and adjustsoutputs on a time scale, implementing the PID control operating in thelength domain requires some additional measures as compared to theconventional PID control. Returning to FIG. 2, an embodiment of thepresent invention comprises a diameter tracking filter 105. In thesimplest form, this filter may be a kinematic model filter. Thoseskilled in the art of control theory and data tracking recognize thatother more elaborate tracking filters, such as a Kalman-Filter, may beused as well in order to produce instantaneous diameter and taper data.The filter receives a signal indicative of a measured diameter from thesensor 15 and a signal indicative of the pull-speed and outputs a signalindicative of the actual diameter. The filter 105 also calculates adiameter change per length (the taper) of the growing ingot from themeasured diameter and the pull-speed and outputs a signal indicative ofthe calculated taper which represents an actual taper of the growingcrystal.

The actual diameter from the filter 105 is evaluated at an adder 106against the target diameter from the diameter profile table 103. Adiameter error, i.e., a difference between the actual diameter and thetarget diameter, is translated into a target taper at a taper profiler107. The taper profiler 107 uses following equations to calculate atarget taper TP_(target).if |TP _(target) |<TP _(limit)TP _(target)=const·Error_(dismeter)otherwiseTP _(target) =+/−TP _(limit)where TP_(target) is the target taper to be achieved, TP_(limit) is themaximum absolute taper (the absolute TP_(target) cannot exceedTP_(limit)), and Error_(diameter) is the difference between the actualdiameter and the target diameter. The target taper TP_(target) is thensupplied to a feed-forward control 108 and a feedback control 109.

The feed-forward control 108 sets, from the target taper TP_(target), aFF adjustment ΔV_(M) to be made to the pull-speed. The control modelaccording to the present invention implements a feed-forward control inaddition to a taper feedback control by setting an exact pull-speedoffset known to be needed to achieve the target taper. The FF adjustmentΔV_(M) is calculated from the empirical data. FIG. 4 shows experimentaldata which indicates how the diameter responded when the pull-speedchanged. In FIG. 4, an ingot was first pulled at the speed of 60 mm/h.When the ingot had grown to 10.5 mm in length, the pull-speed changedfrom 60 mm/h to 120 mm/h. The diameter was being measured and plotted inrelation to the pulled ingot length to monitor its changes during thecrystal growing process. The solid line shows the actual measurementsand the dash line linearly traces the actual measurements. The meniscustapers are calculated from the diameter slopes in the graph in FIG. 4.Since FIG. 4 shows the relationship between a meniscus taper and apull-speed change, it can be determined from FIG. 4 how much thepull-speed should be adjusted to achieve a certain taper change. Sincethe relation between the taper and the pull-speed change can be easilydetermined experimentally, this control model can effect the precise,basic response to diameter deviations and improves its adaptation totuning.

The target taper TP_(target) from the taper profiler 107 is evaluated atan adder 110 in the feedback control 109 against the actual tapercalculated by the diameter tracking filter 105 to determine a tapererror E_(Tpr), or a deviation of the calculated taper from the targettaper TP_(target). In the feedback control 109, the taper error E_(Tpr)is both proportionally and integrally adjusted by a proportionaloperator 111 and an integral operator 112 and translated into a FBadjustment (i-term adjustment and p-term adjustment) to the pull-speedat an adder 113. Please note that in the present invention, the tapererror E_(Tpr) is integrated by the integral operator 112 over the pulledingot length (z), not over time (t), and translated into an adjustmentΔV₀ to the pull-speed (i-term adjustment). The FB adjustment is thenadded at an adder 114 with the FF adjustment ΔV_(M) from thefeed-forward control 108. The target pull-speed from the pull-speedprofile table 102 is adjusted at an adder 115 with a sum of thepull-speed adjustments from the feed-forward control 108 and thefeedback control 109 and supplied to control the servo motor 19. Theadjusted pull-speed is also supplied to the diameter tracking filter105.

As discussed above, the taper error E_(Tpr) is integrated over thepulled ingot length and translated into the i-term adjustment ΔV₀. Thei-term adjustment ΔV₀ is a sum of taper errors over the pulled ingotlength. The difference between ΔV₀ and the target pull-speed is a goodrepresentation of a temperature deviation in the silicon melt 6, evenwhen correcting large diameter errors, provided that the taperlimitation is active. The i-term adjustment ΔV₀ is then integrated by anintegral operator 116 over time and translated into a temperatureadjustment. The temperature adjustment from the i-term adjustment ΔV₀ isadded at an adder 117 with the target temperature from the temperatureprofile table 104 and the measured temperature from the sensor 19 toderive a temperature error. The temperature error goes through the PIDcontroller 118 to control the heather 7.

FIG. 5 shows a comparison of changes of the pull-speed which resultedfrom (1) an implementation of the conventional PID controller and animplementation of the present invention. The pull-speed changes underthe control of the conventional PID controller are shown with brokenlines. The pull-speed changes under the control of the present inventionare shown with solid lines. As shown in FIG. 5, the pull-speed moresignificantly changes under the control of the conventional PIDcontroller than under the control of the present invention. It is clearthat the conventional control repeats overshooting the targets back andforth. These “bumpy” controls also manifest themselves as relativelylarge diameter deviations along the length of an ingot. Compared to theconventional PID control, the control according to the present inventioneffects relatively small diameter deviations along the length while atthe same time the amount of pull-speed variation that is applied inorder to control the diameter is significantly reduced.

FIGS. 6 and 7 also show comparisons between the present invention andthe conventional PID control. FIGS. 6 and 7 show that changes of thediameter and the pull-speed are significantly reduced in the presentinvention compared to those controlled by the conventional PID control.

In the above embodiment, a modified controller is used in which aPI-control operates in the length domain as a taper feed-back controlwith a dynamic taper set-point, or a target taper, derived from adiameter deviation. Such a modified controller has advantages ofimproved control stability and high adaptation for tuning. However,those skilled in the art of control theory can appreciate that thepresent invention is also implementable in a regular PID controlleroperating in the length domain. Such a regular PID controller operatingin the length domain has already been found achieving a significantimprovement over prior art.

As various changes could be made in the above constructions and methodswithout departing from the scope of the invention, it is intended thatall matter contained in the above description or shown in theaccompanying drawings shall be interpreted as illustrative and not in alimiting sense.

1. A method for growing a monocrystalline silicon ingot from siliconmelt according to the CZ process, comprising: defining an error betweena target taper of a meniscus and a measured taper; and translating thetaper error into a feedback adjustment to a pull-speed of the siliconingot, wherein translating the taper error into a feedback adjustmentcomprises integrating the taper error over a length of the grown ingotlength to derive an i-term of the feedback adjustment.
 2. A methodaccording to claim 1, further comprising: defining an error between atarget diameter of the silicon ingot and a measured diameter; andtranslating the diameter error into the target taper.
 3. A methodaccording to claim 2, wherein translating the diameter error into thetarget taper comprises multiplying the diameter error with a constantwithin predetermined taper limits.
 4. A method according to claim 1,wherein the measured taper is calculated from the pull-speed and themeasured diameter.
 5. A method according to claim 1, wherein translatingthe taper error into a feedback adjustment comprises multiplying thetaper error with a constant to derive a p-term of the feedbackadjustment.
 6. A method according to claim 1, further comprisingtranslating the target taper into a feed-forward adjustment to thepull-speed.
 7. A method according to claim 1, further comprisingtranslating the i-term adjustment to the pull-speed into a deviation ofthe temperature of the silicon melt.
 8. A method according to claim 7,wherein translating the i-term adjustment to the pull-speed comprisesintegrating the i-term adjustment over time to derive the temperaturedeviation.
 9. A method according to claim 7, further comprisingeffecting a PID control, using the temperature deviation, to control thetemperature of the silicon melt.
 10. A method according to claim 1,wherein translating the taper error into a feedback adjustment comprisesusing the following equation:$\frac{\partial r}{\partial z} = {{{const} \cdot \Delta}\; v}$ where, rdenotes a diameter of the ingot, z denotes a length of the grown ingotand Δv denotes a pull-speed error.
 11. A method for growing amonocrystalline silicon ingot from silicon melt according to the CZprocess, comprising: defining an error between a target value of aparameter and a measured value of the parameter; and integrating theerror in a length domain to derive a feedback adjustment to a pull-speedof the silicon ingot in order to make a diameter of the silicon ingotuniform along its length.
 12. A method according to claim 11, whereinthe parameter is a taper of an ingot.
 13. A method according to claim11, wherein the parameter is a diameter of the silicon ingot.
 14. Amethod according to claim 11, wherein integrating the error in a lengthdomain comprises integrating the error over a length of the grownsilicon ingot.